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Soccer 1v1- Positioning, Angles, and Dilations

Author:
Lauren Nowak
Topic:
Angles, Area, Dilation, Geometry, Similar Triangles, Geometric Transformations
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While soccer players don't pull out a pencil and paper in the middle of the field, they constantly make mathematical observations and decisions. As they evaluate their position, angle, speed, and next move, they face pressure to make these decisions quickly and accurately! We are going to explore some of the geometric factors affecting decisions that players make on the field, along with some of the advice that is given to players. Because soccer has many moving parts, we're going to start with some simpler scenarios and then build our way into more complicated situations.
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1v1: The Penalty Kick

In a penalty kick, it all comes down to one shooter and the goalkeeper. The shooter has to kick from the penalty spot, and the goalkeeper is not allowed to move until the ball is in motion. Below, explore what would happen if the goalkeeper was allowed to rush forward before the shooter has had a chance to kick. Pay attention to the dilation (enlargement) of the goalkeeper's "shadow" - this represents the amount of the goal he is blocking from the shooter's point of view. Also notice what is happening to the distances between objects and the ratio (fraction or comparison) of these distances.

1v1: Rushing the Penalty Kick

Question #1

Position the goalkeeper so that his shadow exactly covers the goal. Where does he need to position himself so that he is completely blocking the shooter's view? How close is he to the shooter vs. how close is he to the goal?

Question #2

What is the ratio of the goalkeeper's width compared to his shadow? This ratio is sometimes called the zoom factor of a dilation. Why might it be given that name?

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Question #3

Moving up extremely close to a player is often a poor move for a goalkeeper to make (can you think of some reasons why?) If our goalkeeper wanted to move forward so that he still covered half of the goal instead of the entire width, where would he have to be positioned? Relocate him in the applet below and describe where you placed him in the box below.

1v1: Rushing the Penalty Kick Part 2

Question #4

What do you notice about the ratio comparing the goalkeeper's shadow to his original width? How does this connect to his position on the field?

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Question #5

We'd have a pretty boring game of soccer if we could only shoot from in front of the goal. In the applet below, try moving the shooter to different positions, and then moving the Goalkeeper to keep covering half of the goal size. What do you notice about the goalkeeper's position? What do you notice about the ratios from before? Does the angle the shooter makes with the goal affect the ratio?

1v1: Shooting from multiple angles

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Reflection

Watch the two short videos below of goalkeeping tips from soccer coaches. Explain how the explorations we did today connect to their advice for beginner goalies.

Inspired by eldenclessig's applet: https://www.geogebra.org/m/nUv9PNZ3 Image credits: Soccer Ball Close Up https://pixabay.com/photos/football-ball-sport-goal-kick-1678992/ Penalty Kick https://www.flickr.com/photos/fortes/169452422 Goalkeeper https://publicdomainvectors.org/en/free-clipart/Handball-goalkeeper/59996.html Field Alena Paulus / Getty Images Shooter https://freesvg.org/1534935680

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